text: This model for disorder in a crystal starts with the structure factor of a perfect crystal. In one-dimension for simplicity and with N planes, we then start with the expression above for a perfect finite lattice, and then this disorder only changes  by a multiplicative factor, to give
  where the disorder is measured by the mean-square displacement of the positions  from their positions in a perfect one-dimensional lattice:  , i.e., , where  is a small (much less than ) random displacement. For disorder of the first kind, each random displacement  is independent of the others, and with respect to a perfect lattice. Thus the displacements  do not destroy the translational order of the crystal. This has the consequence that for infinite crystals () the structure factor still has delta-function Bragg peaks – the peak width still goes to zero as , with this kind of disorder. However, it does reduce the amplitude of the peaks, and due to the factor of  in the exponential factor, it reduces peaks at large  much more than peaks at small .
  The structure is simply reduced by a  and disorder dependent term because all disorder of the first-kind does is smear out the scattering planes, effectively reducing the form factor.
  In three dimensions the effect is the same, the structure is again reduced by a multiplicative factor, and this factor is often called the Debye–Waller factor. Note that the Debye–Waller factor is often ascribed to thermal motion, i.e., the  are due to thermal motion, but any random displacements about a perfect lattice, not just thermal ones, will contribute to the Debye–Waller factor.